Localization and Affine Schemes over $\mathbb{F}_1$

We develop the basic notions of commutative algebra and algebraic geometry over the field with one element $\mathbb{F}_1$, working within the Connes-Consani framework, which models $\mathbb{F}_1$-algebras as monoid objects in the category of $\Gamma$-sets. In this setting, $\mathbb{F}_1$-algebras generalize commutative rings by encoding the algebraic structure functorially, using machinery originating in homotopy theory. Our main contribution is a theory of localization for $\mathbb{F}_1$-algebras and the construction of prime spectrum $\Spec A$ for a commutative $\mathbb{F}_1$-algebra $A$. We then prove that $\Gamma(X, \mathcal{O}_X) = A$ for any absolute affine scheme $X=\Spec A$ and establish an anti-equivalence between the category of commutative $\mathbb{F}_1$-algebras and the category of absolute affine schemes.